I'm looking to find a formula for solving for a growth constant, given a known start, end and number of periods.
Best illustrated with an example:
Tom wants to do 10,000 pushups this year. He'll start on day 1 with 1 pushup, and increase X amount each day until he's hit 10,000 for the year.
The increase each day should be constant (e.g., Tom adds 3 pushups a day).
Is there a formula for determining this growth constant?
Thanks!
Let $x$ be the additional amount of push-ups done each day. Then, after $N$ days, a total of $$ \sum_{k=0}^{N}(1+kx) $$ push-ups have been done. We can simplify this sum as follows:
$$ \sum_{k=0}^{N}(1+kx)=\sum_{k=0}^{N}1+x\sum_{k=0}^{N}k=N+x\cdot\frac{N(N+1)}{2}. $$
Now, suppose you want $M$ push-ups done in total. This means you want
$$ M=N+x\cdot\frac{N(N+1)}{2}\qquad\qquad(*) $$
Thus, whenever you have two of the three variables $x,N,M$ given, you can solve for the third from $(*)$ as follows:
For example, if you want to do $10000$ push-ups in $365$ days, you have $M=10000$ and $N=365$. Hence, by $(*)$,
$$ 10000=365+x66795\implies x=\frac{9635}{66795}\approx 0.144. $$
Thus, you should do $0.144$ more push-ups each day. Since $1/0.144\approx 7$, you could add an additional push-up once every $7$ days.